There aren’t any formal categories of game engines really, so I’m just going to make up some groupings and put some engines I know about into them.
Major Studio Engines.
Prominent examples include Frostbite (EA/Battlefront), RAGE (Rockstar/GTA) , Anvil (Ubisoft/ Assassin’s Creed).
These are the big AAA game engines that outsiders don’t get to use. They do lots of specialized things and the results are very very pretty. The tools themselves are often more powerful, less flexible, and have less UX polish than the next category.
Top Tier Licensed 3d Engines.
I only really put two engines here, Unity and Unreal. These engines are free to get your hands on for learning and between them can deploy to nearly every platform and can make just about any type of game.
Mid Tier Licensed 3d Engines
This is a big field with a lot of variety. These include CryEngine, high end but a pain and not flexible enough and Godot which is up and coming in popularity but still too new and without the community and battle testing needed to get to top tier.
These get their own category because they’re doing something that neither Unity or Unreal is doing well, putting 3d content in a mobile browser. I’m most familiar with Playcanvas, which looks a lot like Unity in a browser window.